The Stability Of Differential Equations With Time-Delay

In 1892, Russian mathematician, mechanician and physicist, Lyapunov, proposed the mathematics definition of the stability of motion and its general research method in his doctoral dissertation"The general problem of the stability of motion",which established the foundation of the stability theory. With scientific progress and social development, Lyapunov theory is being expanded and developed constantly and is applied extensively in delay differential equations. Because of the technology difficulties,we can't study the stability of all variables, the limitation of practical conditions, or it is not necessary to study all variables considering the actual need. As a result ,studying the partial stability of differential equations becomes more important. In addition, the partial stability of differential equations is widely used in science and technology.The aim of the dissertation is to investigate the stability(including Lyapunov stability and Lipschitz stability) and the partial stability, double stability of time-delay differential equations and neutral systems. The author uses the method of differential inequalities with time-delay and integral inequalities to establish some all variables stability criteria and partial stability criteria and double stability criteria.This thesis has six sections.The prologue first introduces the current research situation of the all variables stability and the partial stability, double stability of delay differential equation, and then emphasizes the significance of our work in the thesis.In first chapter, the problems are posed.In second chapter, basic knowledge. It mainly introduces the definition of stability and the partial stability, double stability for delay differential equations. And some lemmas are presented.In the third chapter, the author develops an integral inequality in [1], and uses the conclusion to establish the stability to the nonlinear delay differential equations.In the fourth chapter, we develop a differential inequality with time-delay in [2], and establish the double stability to the nonlinear delay differential equations and stability to nonlinear neutral systems with multiple delay.In the fifth chapter, we continue developing the integral inequality in [15], and establish double stability to the nonlinear neutral differential equations.